Generalized jordan curve theorem

In summary, the conversation discusses the possibility of using the proof of the generalized Jordan curve theorem, found on page 169 of Hatcher's algebraic topology book, to show that a general (n-1)-manifold divides S^n into two components. It is mentioned that the complement of an n-1-sphere in S^n has the homology of S^0, but this does not hold for general n-1-manifolds. The issue is that the corresponding terms of the Mayer-vietoris sequence used in Hatcher's proof are not zero where they need to be. The use of Alexander duality is suggested as a possible solution.
  • #1
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Is anyone here familiar with the proof (using homology) of the generalized Jordan curve theorem, that a subspace of S^n homeomorphic to S^(n-1) divides it into two components? It can be found on page 169 of Hatcher's algebraic topology book, which can be downloaded from http://www.math.cornell.edu/~hatcher/AT/ATpage.html" [Broken] page.

I'm wondering if it's possible to use the same kind of proof to show that a general (n-1)-manifold divides S^n into two components. It was shown that the complement of an n-1-sphere in S^n actually has the homology of S^0, which is much stronger, and won't hold for general n-1-manifolds. But all I need is that the 0th reduced homology group is Z, which does seem to be true. The problem is that the corresponding terms of the Mayer-vietoris sequence used in Hatcher's proof aren't zero where they need to be. Does anyone have any ideas?
 
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  • #2
are you using alexander duality? see p.254.
 
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  • #3
I haven't gotten to that yet (Im just starting cohomology), though I see how it would follow from there. Is there an easier method, or a way to extract that bit of information using the idea of alexander duality without developing all the machinery?
 

1. What is the Generalized Jordan Curve Theorem?

The Generalized Jordan Curve Theorem is a mathematical theorem that states that any loop in a plane can be continuously deformed into a single point without breaking or self-intersecting, as long as it is a simple closed curve. This means that any closed shape in a plane can be shrunk down to a single point without tearing or overlapping itself.

2. What is the difference between the Generalized Jordan Curve Theorem and the Jordan Curve Theorem?

The Jordan Curve Theorem is a special case of the Generalized Jordan Curve Theorem, where the loop in question is a simple closed curve. This means that the loop does not intersect itself at any point. The Generalized Jordan Curve Theorem extends this concept to any closed curve, even if it is not simple.

3. What are some real-world applications of the Generalized Jordan Curve Theorem?

The Generalized Jordan Curve Theorem has many applications in various fields such as topology, geometry, and computer science. It is used in image processing and computer graphics to identify and manipulate shapes, in robotics for path planning and obstacle avoidance, and in geography for understanding the shapes of coastlines and rivers.

4. How was the Generalized Jordan Curve Theorem proven?

The Generalized Jordan Curve Theorem was first proven in 1905 by the French mathematician Camille Jordan. However, his proof was later found to be incomplete. It was not until 1920 when the American mathematician Oswald Veblen provided a complete proof of the theorem.

5. Are there any exceptions to the Generalized Jordan Curve Theorem?

Yes, there are a few exceptions to the Generalized Jordan Curve Theorem. One example is the Alexander horned sphere, which is a three-dimensional shape that cannot be continuously deformed into a single point without self-intersecting. This exception highlights the importance of the simple closed curve requirement in the theorem.

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